The Mangasarian-Fromovitz constraint qualification (MFCQ) condition.
Consider the case where \(C = \lbrace  w\in \mathbb {R}^d: g(w) \ge 0\rbrace \) , for a smooth function \(g: \mathbb {R}^d \rightarrow \mathbb {R}^{n_g}\) . Denote \(\operatorname{I}(w) = \lbrace  i \in \lbrace 1,\dots ,n_g\rbrace , g_i(w) \le 0\rbrace \)  as the set of active constraints. We say that the MFCQ condition holds at \(w\in \mathbb {R}^d\)  if there exists \(v \in \mathbb {R}^d\)  such that \(\nabla g_i(w)^{\top }v \ge 0\)  for all \(i \in \operatorname{I}(w)\) . If the MFCQ condition holds at \(w\in C\) , then we can write down \(T_C(w)= \lbrace  v \in \mathbb {R}^d, \nabla g_i(w)^\top v \ge 0, \text{for all} \, i \in \operatorname{I}(w) \rbrace \) 
and
\(N_C(w) = \lbrace  -\sum _{i=1}^{n_g} \lambda _i \nabla g_i(w), \lambda _i \in \mathbb {R}_+ \textrm { and } \lambda _i=0 \textrm { if } i \notin \operatorname{I}(w) \rbrace \)  (see, e.g., [1]}). We might notice here, that in the context of Theorem REF  the MFCQ condition holds at every \(w\in C_{\epsilon }\) .
